Estimating the local mean for Bayesian maximum entropy by generalized least squares and maximum likelihood, and an application to the spatial analysis of a censored soil variable

The Bayesian maximum entropy (BME) method is a valuable tool, with rigorous theoretical underpinnings, with which to predict with soft (imprecise) data. The methodology uses a general knowledge base to derive a joint prior distribution of the data and the prediction by the criterion of maximum entropy; the hard (precise) and soft data are then processed using this prior distribution to yield a posterior distribution that provides the BME prediction. The general knowledge base commonly consists of the mean and covariance functions, which may be extracted from the data. The common method for extracting the mean function from the data is a generalized least squares (GLS) approach. However, when the soft data take the form of intervals of plausible values, this method can result in errors in the BME predictions. This paper suggests a maximum likelihood (ML) method for fitting the local mean. The two methods are compared in terms of their predictions, firstly on simulated random fields and then on a case study to predict the depth of soil using some censored data. The results show that the ML method can result in more accurate BME predictions; the degree of improvement over the GLS method depends on the parameters of the spatial covariance model.